Over the last years it has been discovered that surprisingly many different problems of Analysis naturally lead to questions about singularities in (vector) measures (that is, parts of the measure that are not absolutely continuous with respect to Lebesgue measure). These problems come from both "pure" Analysis, such as the question for which measures Rademacher's Theorem on the differentiability of Lipschitz functions holds, as well as "applied" Analysis, such as the question to determine the fine structure of slip lines in elasto-plasticity.
It's a remarkable fact that many of the (vector) measures that naturally occur in the wild, satisfy an (under-determined) PDE and thus the task arises to analyse these PDEs and in particular their possible singularities. This leads to restrictions on the shape of these singularities, which yields several interesting results in various areas of Analysis. The essential difficulty for the analysis of these measure-PDEs is that many standard methods (such as harmonic analysis) are much weaker in an L^1-context and thus new strategies need to be developed.
In this talk, which is aimed at a general mathematical audience, I will survey recent (and ongoing) work on several of these questions.